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Jun 30, 2018
06/18

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Kazuyuki Fujii

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In this comment we give a simple analytic approximate solution to the infectious recovery SIR (irSIR) model given by J. Cannarella and J. A. Spechler arXiv:1401.4208, which is a variant of the traditional SIR model.

Topics: Physics, Physics and Society

Source: http://arxiv.org/abs/1402.1225

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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We in this paper consider a further generalization of the (optical) holonomic quantum computation proposed by Zanardi and Rasetti (quant-ph/9904011), and reinforced by Fujii (quant-ph/9910069) and Pachos and Chountasis (quant-ph/9912093). We construct a quantum computational bundle on some parameter space, and calculate non-abelian Berry connections and curvatures explicitly in the special cases. Our main tool is unitary coherent operators based on Lie algebras su(n+1) and su(n,1), where the...

Source: http://arxiv.org/abs/quant-ph/0005129v1

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Sep 20, 2013
09/13

by
Kazuyuki Fujii

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In this paper we point out that the Jaynes-Cummings model without taking a renonance conditon gives a non-commutative version of the simple spin model (including the parameters $x$, $y$ and $z$) treated by M. V. Berry. This model is different from usual non-commutative ones because the x-y coordinates are quantized, while the z coordinate is not. One of new and interesting points in our non-commutative model is that the strings corresponding to Dirac ones in the Berry model exist only in states...

Source: http://arxiv.org/abs/quant-ph/0410201v2

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In this paper we present a new algebraic structure (a super hyperbolic system in our terminology) for finite quantum systems, which is a generalization of the usual one in the two-level system. It fits into the so-called generalized Pauli matrices, so they play an important role in the theory. Some deep relation to the modified Bessel functions of integer order is pointed out. By taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and...

Source: http://arxiv.org/abs/0704.1844v2

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Sep 20, 2013
09/13

by
Kazuyuki Fujii

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This is a challenging paper including some review and new results. Since the non-commutative version of the classical system based on the compact group SU(2) has been constructed in (quant-ph/0502174) by making use of Jaynes-Commings model and so-called Quantum Diagonalization Method in (quant-ph/0502147), we construct a non-commutative version of the classical system based on the non-compact group SU(1,1) by modifying the compact case. In this model the Hamiltonian is not hermite but pseudo...

Source: http://arxiv.org/abs/quant-ph/0506026v2

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Sep 19, 2013
09/13

by
Kazuyuki Fujii

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In this paper we consider a quantum open system and treat the master equation with some restricted dissipator which consists of a set of projection operators (projectors). The exact solution is given under the commutable approximation (in our terminology). This is the first step for constructing a general solution.

Source: http://arxiv.org/abs/0708.4047v1

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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We give a modern approach to the famous Cardano and Ferrari formulas in the algebraic equations with three and four degrees. Namely, we reconstruct these formulas from the point of view of superposition principle in quantum computation based on three and four level systems which are being developed by the author. We also present a problem on some relation between Galois theory and Qudit theory.

Source: http://arxiv.org/abs/quant-ph/0311102v2

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In the preceding paper quant-ph/0312060 we considered a general model of an atom with n energy levels interacting with n-1 external laser fields and constructed a Rabi oscillation in the case of n =3, 4 and 5. In the paper we present a systematic method getting along with computer to construct a Rabi oscillation in the general case.

Source: http://arxiv.org/abs/quant-ph/0605132v1

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Sep 20, 2013
09/13

by
Kazuyuki Fujii

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The quantum damped harmonic oscillator is described by the master equation with usual Lindblad form. The equation has been solved completely by us in arXiv : 0710.2724 [quant-ph]. To construct the general solution a few facts of representation theory based on the Lie algebra $su(1,1)$ were used. In this paper we treat a general model described by a master equation with generalized Lindblad form. Then we examine the algebraic structure related to some Lie algebras and construct the interesting...

Source: http://arxiv.org/abs/0802.3252v1

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Sep 17, 2013
09/13

by
Kazuyuki Fujii

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In this paper we propose a Hamiltonian generalizing the interaction of the two--level atom and both the single radiation mode and external field $...$ a kind of cavity QED. We solve the Schrodinger equation in the strong coupling regime by making use of rotating wave approximation under new resonance conditions containing the Bessel functions and etc, and obtain unitary transformations of four types corresponding to Rabi oscillations which perform quantum logic gates in Quantum Computation.

Source: http://arxiv.org/abs/quant-ph/0305155v1

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Sep 17, 2013
09/13

by
Kazuyuki Fujii

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In the first half we make a short review of coherent states and generalized coherent ones based on Lie algebras su(2) and su(1,1), and the Schwinger's boson method to construct representations of the Lie algebras. In the second half we make a review of recent developments on both swap of coherent states and cloning of coherent states which are important subjects in Quantum Information Theory.

Source: http://arxiv.org/abs/quant-ph/0207178v1

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In this paper we construct a non-commutative version of the Hopf bundle by making use of Jaynes-Commings model and so-called Quantum Diagonalization Method. The bundle has a kind of Dirac strings. However, they appear in only states containing the ground one (${\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F} \subset {\cal F}\times {\cal F}$) and don't appear in remaining excited states. This means that classical singularities are not universal in the process of non-commutativization....

Source: http://arxiv.org/abs/quant-ph/0502174v2

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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A volume form $H$ on the $n$--dimensional sphere $S^n$ is closed $(dH=0)$, so that it is locally written as $H=dB$, where B is a $(n-1)$--form. In the first half we give an explicit form to B and, moreover, a speculation concerning higher order U(1) bundles. In the second half we apply our B in the case $n=3$ to a string soliton solution discussed by Perry and Schwarz.

Source: http://arxiv.org/abs/hep-th/9711018v1

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Jul 20, 2013
07/13

by
Kazuyuki Fujii

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We give a possible generalization to the example in the paper of Zanardi and Rasetti (quant-ph/9904011). For this generalized one explicit forms of adiabatic connection, curvature and etc. are given.

Source: http://arxiv.org/abs/quant-ph/9910069v1

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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In this paper we generalize the Jaynes--Cummings Hamiltonian by making use of some operators based on Lie algebras su(1,1) and su(2), and study a mathematical structure of Rabi floppings of these models in the strong coupling regime. We show that Rabi frequencies are given by matrix elements of generalized coherent operators (quant--ph/0202081) under the rotating--wave approximation. In the first half we make a general review of coherent operators and generalized coherent ones based on Lie...

Source: http://arxiv.org/abs/quant-ph/0203135v6

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Sep 17, 2013
09/13

by
Kazuyuki Fujii

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In this paper we present a non-Gaussian integral based on a cubic polynomial, instead of a quadratic, and give a fundamental formula in terms of its discriminant. It gives a mathematical reinforcement to the recent result by Morozov and Shakirov. We also present some related results. This is simply one modest step to go beyond the Gaussian but it already reveals many obstacles related with the big challenge of going further beyond the Gaussian.

Source: http://arxiv.org/abs/0912.2135v3

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Jul 20, 2013
07/13

by
Kazuyuki Fujii

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The aim of this paper is to introduce our idea of Holonomic Quantum Computation (Computer). Our model is based on both harmonic oscillators and non-linear quantum optics, not on spins of usual quantum computation and our method is moreover completely geometrical. We hope that therefore our model may be strong for decoherence.

Source: http://arxiv.org/abs/quant-ph/0107128v1

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Sep 17, 2013
09/13

by
Kazuyuki Fujii

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First we make a brief review of coherent states and prove that the resolution of unity can be obtained by the 1-st Chern character of some bundle. Next we define a Grassmann manifold for a set of coherent states and construct the pull-back bundle making use of a projector from the parameter space to this Grassmann manifold. We study some geometric properties (Chern-characters mainly) of these bundles. Although the calculations of Chern-characters are in general not easy, we can perform them for...

Source: http://arxiv.org/abs/hep-ph/0108219v1

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In the first half we show an interesting relation between coherent states and the Bell states in the case of spin 1/2, which was suggested by Fivel. In the latter half we treat generalized coherent states and try to generalize this relation to get several generalized Bell states. Our method is based on a geometry and our task may give a hint to open a deep relation between a coherence and an entanglement.

Source: http://arxiv.org/abs/quant-ph/0105077v2

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Jul 20, 2013
07/13

by
Kazuyuki Fujii

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This is a continuation of the paper (quant-ph/0009012). In this letter we extend coherent operators and study some basic properties (the disentangling formula, resolution of unity, commutation relation, etc). We also propose a perspective of our work.

Source: http://arxiv.org/abs/quant-ph/0009116v2

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Sep 22, 2013
09/13

by
Kazuyuki Fujii

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The Simpson's formula is obtained by approximating the integral of a function on some interval by the integral of the quadratic polynomial determined by the function. However, a multidimensional analogue of the formula has not been given as far as we know. In this paper such a formula is given. Our formula is simple and beautiful, so it may be convenient in Mathematics or Mathematical Physics.

Source: http://arxiv.org/abs/0908.3533v1

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Sep 22, 2013
09/13

by
Kazuyuki Fujii

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A set of generators of generalized Pauli matrices play a crucial role in quantum computation based on n level systems of an atom. In this paper we show how to construct them by making use of Rabi oscillations. We also construct the generalized Walsh-Hadamard matrix in the case of three level systems and present some related problems.

Source: http://arxiv.org/abs/quant-ph/0309132v1

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Sep 19, 2013
09/13

by
Kazuyuki Fujii

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In this paper we show how to calculate explicitly the exponential of certain matrices, which are evolution operators governing the interaction of the four level system of atoms and the radiation, etc. We present a consistent method in terms of the magic matrix by Makhlin. As a closely related subject, we derive a closed form expression of the Baker-Campbell-Hausdorff formula for a class of matrices in SU(4), by use of the method developed by the present authors in quant-ph/0610009.

Source: http://arxiv.org/abs/quant-ph/0701092v3

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Sep 23, 2013
09/13

by
Kazuyuki Fujii

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This is a continuation of the preceding paper (hep-ph/0108219). First of all we make a brief review of generalized coherent states based on Lie algebra su(1,1) and prove that the resolution of unity can be obtained by the curvature form of some bundle. Next for a set of generalized coherent states we define a universal bundle over the infinite-dimensional Grassmann manifold and construct the pull-back bundle making use of a projector from the parameter space to this Grassmann one. We mainly...

Source: http://arxiv.org/abs/hep-th/0109215v1

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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In this note I introduce a mysterious approximation called the rotating wave approximation (RWA) to undergraduates or non-experts who are interested in both Mathematics and Quantum Optics. In Quantum Optics it plays a very important role to obtain an analytic approximate solution of some Schr{\"o}dinger equation, while it is curious from the mathematical point of view. I explain it carefully with two coherent oscillations for them and expect that they will overcome the problem in the near...

Source: http://arxiv.org/abs/1301.3585v2

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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In this paper a higher order non-linear differential equation is given and it becomes a higher order Airy equation (in our terminology) under the Cole-Hopf transformation. For the even case a solution is explicitly constructed, which is a generalization of the Airy function.

Source: http://arxiv.org/abs/0712.2481v1

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Sep 19, 2013
09/13

by
Kazuyuki Fujii

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In this paper we propose some Hamiltonian characterizing the interaction of the two-level atom and both the single radiation mode and external field, which might be a generalization of that of Sch{\"o}n and Cirac (quant-ph/0212068). We solve them in the strong coupling regime under some conditions (the rotating wave approximation, resonance condition and etc), and obtain unitary transformations of four types to perform Quantum Computation.

Source: http://arxiv.org/abs/quant-ph/0303118v1

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Sep 23, 2013
09/13

by
Kazuyuki Fujii

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In this note an explicit expression of $\exp A\ (A\in so(4))$ is given in terms of the magic matrix by Makhlin.

Source: http://arxiv.org/abs/1111.1487v1

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Sep 22, 2013
09/13

by
Kazuyuki Fujii

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We make a brief comment on measurement of quantum operators with degenerate eigenstates and apply to quantum teleportation. We also try extending the quantum teleportation by Bennett et al [5] to more general situation by making use of generalized Bell states.

Source: http://arxiv.org/abs/quant-ph/0106018v2

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Sep 22, 2013
09/13

by
Kazuyuki Fujii

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In (quant-ph/0701141) Rajeev studied quantization of the damped simple harmonic oscillator and introduced a complex-valued Hamiltonian (which is normal). In this note we point out that the quantization is interpreted as a quantum mechanics with {\bf complex time}. We also present a problem on quantization of classical control systems.

Source: http://arxiv.org/abs/quant-ph/0702148v1

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Sep 22, 2013
09/13

by
Kazuyuki Fujii

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We make a brief review of (optical) Holonomic Quantum Computer (or Computation) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093), and give a mathematical reinforcement to their works.

Source: http://arxiv.org/abs/quant-ph/0004102v1

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Sep 23, 2013
09/13

by
Kazuyuki Fujii

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In this note a generalization of the Lamb-Bateman integral equation is presented and its solution is given in terms of {\bf fractional derivatives}. This is a comment one to the paper by Babusci, Dattoli and Sacchetti (arXiv:1006.0184 [math-ph]).

Source: http://arxiv.org/abs/1006.2424v2

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In this note we make a comment on the paper by Jarlskog (math-ph/0504049) in which she gave a parametrisation to unitary matrices. Namely, we show that her recursive method is essentially obtained by the canonical coordinate of the second kind in the Lie group theory, and propose a problem on constructing unitary gates in quantum computation by making use of the parametrisation.

Source: http://arxiv.org/abs/math-ph/0505047v3

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Jul 20, 2013
07/13

by
Kazuyuki Fujii

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The purpose of this paper is both to provide mathematical reinforcements to the paper [Mecozzi and Bellini : arXiv:1110.1253 [hep-ph]] by taking decoherence into consideration and to present some important problems related. We claim that neutrinos have superluminality as a latent possibility.

Source: http://arxiv.org/abs/1203.6425v2

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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In this paper we treat a non-Gaussian integral and give a fundamental formula in terms of discriminant. We also present some related problems. This is a comment paper to arXiv:0903.2595 [math-ph] by Morozov and Shakirov.

Source: http://arxiv.org/abs/0905.1363v3

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In this note we make a short review of constructions of n-repeated controlled unitary gates in quantum logic gates.

Source: http://arxiv.org/abs/quant-ph/0101054v1

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In this chapter we treat the quantum damped harmonic oscillator, and study mathematical structure of the model, and construct general solution with any initial condition, and give a quantum counterpart in the case of taking coherent state as an initial condition. This is a simple and good model of Quantum Mechanics with dissipation which is important to understand real world, and readers will get a powerful weapon for Quantum Physics.

Source: http://arxiv.org/abs/1209.1437v1

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In this paper we consider some (generalized) models which deal with n--level atom interacting with a single radiation mode and study a general structure of Rabi floppings of these generalized ones in the strong coupling regime. To solve these models we introduce multi cat states of Schr{\" o}dinger in our terminology and extend the result in (quant--ph/0203135) which Rabi frequencies are given by matrix elements of generalized coherent operators (quant--ph/0202081) under the rotating--wave...

Source: http://arxiv.org/abs/quant-ph/0210166v2

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Sep 17, 2013
09/13

by
Kazuyuki Fujii

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We construct the exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three "reverse" gates. This is a natural extension of the qubit case. We also consider a similar subject on the Fock space, but in this case we meet with some different situation. However we can construct the exchange gate by making use of generalized coherent operator based on the Lie algebra su(2) which is a well--known method in Quantum Optics. We...

Source: http://arxiv.org/abs/quant-ph/0207002v3

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Jul 20, 2013
07/13

by
Kazuyuki Fujii

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In this letter we make a brief review of some basic properties (the matrix elements, the trace, the Glauber formula) of coherent operators and study the corresponding ones for generalized coherent operators based on Lie algebra su(1,1). We also propose some problems.

Source: http://arxiv.org/abs/quant-ph/0009012v2

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Sep 20, 2013
09/13

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Kazuyuki Fujii

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In this paper we treat the 2--level system interacting with external fields without the rotating wave approximation and construct some approximate solutions in the strong coupling regime.

Source: http://arxiv.org/abs/quant-ph/0301145v2

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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In this paper we give a geometric parametrization to the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix and the Jarlskog invariant, which is based on two flag manifolds $SU(3)/U(1)^{2}$. To treat a fourth generation of quarks on CP violation we generalize the parametrization to one based on two flag manifolds $SU(4)/U(1)^{3}$.

Source: http://arxiv.org/abs/0901.2180v2

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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In the preceding paper arXiv:0802.3252 [quant-ph] we treated a model given by a master equation with generalized Lindblad form, and examined the algebraic structure related to some Lie algebras and constructed an approximate solution. In this paper we apply a unitary transformation by the squeezing operator to the master equation. Then the generalized Lindblad form is tranformed to the usual Lindblad one, while the (original) Hamiltonian is tranformed to somewhat complicated one. As a result we...

Source: http://arxiv.org/abs/0803.3105v1

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Sep 22, 2013
09/13

by
Kazuyuki Fujii

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In this paper we study the behavior of (laser--cooled) m--atoms trapped in a cavity interacting with a photon $...$ Cavity QED $...$ and attempt to solve the Schr{\"o}dinger equation of this model in the strong coupling regime. In the case of m = 2 we construct Bell--Schr{\" o}dinger cat states (in our terminology) and obtain with such bases some unitary transformations by making use of the rotating wave approximation under new resonance conditionscontaining the Bessel functions,...

Source: http://arxiv.org/abs/quant-ph/0307168v1

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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Explicit forms are given of matrix elements of generalized coherent operators based on Lie algebras su(1,1) and su(2). We also give a kind of factorization formula of the associated Laguerre polynomials.

Source: http://arxiv.org/abs/quant-ph/0202081v4

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Sep 24, 2013
09/13

by
Kazuyuki Fujii

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Geometrical aspects of quantum computing are reviewed elementarily for non-experts and/or graduate students who are interested in both Geometry and Quantum Computation. In the first half we show how to treat Grassmann manifolds which are very important examples of manifolds in Mathematics and Physics. Some of their applications to Quantum Computation and its efficiency problems are shown in the second half. An interesting current topic of Holonomic Quantum Computation is also covered. In the...

Source: http://arxiv.org/abs/quant-ph/0103011v5

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Jun 30, 2018
06/18

by
Kazuyuki Fujii

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In this paper we discuss a master equation applied to the two level system of an atom and derive an exact solution to it in an abstract manner. We also present a problem and a conjecture based on the three level system. Our results may give a small hint to understand the huge transition from Quantum World to Classical World. To the best of our knowledge this is the finest method up to the present.

Topics: Quantum Physics, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1405.2604

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Sep 19, 2013
09/13

by
Kazuyuki Fujii

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In this paper we propose a Hamiltonian of the n-level system by making use of generalized Pauli matrices.

Source: http://arxiv.org/abs/quant-ph/0302050v2

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Sep 21, 2013
09/13

by
Kazuyuki Fujii

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In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on $SU(3)$ and its dimension is $8$, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator.

Source: http://arxiv.org/abs/1008.4624v1

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Sep 18, 2013
09/13

by
Kazuyuki Fujii

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This is a sequel to the papers (quant-ph/9910063) and (quant-ph/0004102). The aim of this paper is to give mathematical foundations to Holonomic Quantum Computation (Computer) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093). In 2-qubit case we give an explicit form to non-abelian Berry connection of quantum computational bundle which is associated with Holonomic Quantum Computation, on some parameter space. We also suggest a possibility that not...

Source: http://arxiv.org/abs/quant-ph/0101102v1